Summary of Double Integrals over General Regions

A general bounded region $D$ on the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region.
To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.
We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.
We can use Fubini’s theorem for improper integrals to evaluate some types of improper integrals.

Key Equations

Iterated integral over a Type I region
$\displaystyle\iint_{D} f(x,y)dA=\displaystyle\iint_{D} f(x,y)dydx=\displaystyle\int_{a}^{b}\left[\displaystyle\int_{g_{1}(x)}^{g_{2}(x)}f(x,y)dy\right] dx$
Iterated integral over a Type II region
$\displaystyle\iint_{D} f(x,y)dA=\displaystyle\iint_{D} f(x,y)dydx=\displaystyle\int_{c}^{d}\left[\displaystyle\int_{h_{1}(y)}^{h_{2}(y)}f(x,y)dx\right] dy$

improper double integral: a double integral over an unbounded region or of an unbounded function

Type I: a region $\bf{D}$ in the $xy$ -plane is Type I if it lies between two vertical lines and the graphs of two continuous functions $g_{1}(x)$ and $g_{2}(x)$

Type II: a region $\bf{D}$ in the $xy$ -plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions $h_{1}(y)$ and $h_{2}(y)$